Download Algebra 1 Summer Institute 2014 The Fair/Unfair Polarization

Algebra 1 Summer Institute 2014
The Fair/Unfair Polarization
Summary
Goals
Participant Handouts
In this activity, participants
 Explore finite, equally
will determine
likely probability models
mathematical and
 Determine mathematical
experimental probabilities
probabilities and the
and compare them to
probability table
determine if a game is fair
or unfair.
Materials
Technology
Source
Paper
Dice
LCD Projector
Facilitator Laptop
NCTM
1. The ESP Polarization
Estimated Time
90 minutes
Mathematics Standards
Common Core State Standards for Mathematics
MAFS. 7.SP.3: Investigate chance processes and develop, use, and evaluate probability
models
3.5: Understand that the probability of a chance event is a number between 0 and 1
that expresses the likelihood of the event occurring. Larger numbers indicate
greater likelihood. A probability near 0 indicates an unlikely event, a probability
around ½ indicates an event that is neither unlikely nor likely, and a probability
near 1 indicates a likely event.
3.6: Approximate the probability of a chance event by collecting data on the chance
process that produces it and observing its long-run relative frequency, and predict
the approximate relative frequency given the probability. For example, when
rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200
times, but probably not exactly 200 times.
3.7: Develop a probability model and use it to find probabilities of events. Compare
probabilities from a model to observed frequencies; if the agreement is not good,
explain possible sources of the discrepancy.
a. Develop a uniform probability model by assigning equal probability to all
outcomes, and use the model to determine probabilities of events. For
example, if a student is selected a random from a class, find the probability
that Jane will be selected and the probability that a girl will be selected.
b. Develop a probability model (which may not be uniform) by observing
frequencies in data generated from a chance process. For example, find the
approximate probability that a spinning penny will land heads up or that a
tossed paper clip will land open end down. Do the outcomes for the spinning
penny appear to be equally likely based on the observed frequencies.
3.8: Find the probabilities of compound events using organized lists, tables, trees, and
simulation.
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Algebra 1 Summer Institute 2014
a. Understand that, just as with simple events, the probability of a compound
event is the fraction of outcomes in the sample space for which the compound
event occurs.
b. Represent sample spaces for the compound events using methods such as
organized lists, tables and tree diagrams. For an event described in everyday
language (e.g.,”rolling double sixes”), identify the outcomes in the sample
space which compose the event.
c. Design and use a simulation to generate frequencies for compound events. For
example, use random digits as a simulation tool to approximate the answer to
the question: If 40% of donors have type A blood, what is the probability that
it will take at least 4 donors to find one with type A blood?
MAFS.912.S-IC.1: Understand and evaluate random processes underlying statistical
experiments
1.1: Understand statistics as a process for making inferences about population
parameters based on a random sample from that population.
1.2: Decide if a specified model is consistent with results from a given datagenerating process, e.g., using simulation. For example, a model says that a
spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row
cause you to question the model?
Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use tools appropriately
Instructional Plan
The next activity is a game of chance for two players using two dice of different colors
(one red, one blue).
1. First, what are the mathematical probabilities for a fair die? (Slide 2)
Face Frequency Probability
1
1
1/6
2
1
1/6
3
1
1/6
4
1
1/6
5
1
1/6
6
1
1/6
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Algebra 1 Summer Institute 2014
2. Without them knowing, give them a loaded or unfair die and ask them to roll it 30
times and count the number of times each face shows up.
After they figure out that the die is not fair, give them a fair one so they can repeat
the experiment. Pool the results from all participants. How close are the
experimental probabilities to the mathematical model?
Suppose you roll a die three times, and the die comes up with a 5 all three times.
What is the probability that the fourth roll will be a 5?
3. Back to the game with two dice. Each of the two players rolls a die, and the
winner is determine by the sum of the faces: (Slide 3)


Player A wins when the sum is 2, 3, 4, 10, 11, or 12
Player B wins when the sum is 5, 6, 7, 8, or 9.
Use your own colored dice to collect data as we play the game.
If this game is played many times, which player do you think will win more often,
and why?
For now, let their instincts guide their answer. Later on we'll analyze this problem
more thoroughly.
Many people select Player A, since there are more outcomes that will cause this
player to win. But in order to be sure, we need to determine the mathematical
probability for each player winning. One way to arrive at these mathematical
probabilities is to describe all possible outcomes when you toss a pair of dice and
compute the sum of their faces.
4. To analyze this problem effectively, we need a clear enumeration of all possible
outcomes. Let's examine one scheme that is based on a familiar idea: an addition
table.
Start with a two-dimensional table:
Blue
Die
+
1
2
3
4
5
6
1
Red Die
2 3 4 5
6
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Algebra 1 Summer Institute 2014
Ask participants to complete the table (be aware of the difference between such
outcomes as 2 + 4 and 4 + 2 since they come from different dice. How many
entries does the table have?
5. For how many of the 36 outcomes will Player A win? For how many of the 36
outcomes will Player B win? Who is more likely to win this game?
6. Change the rules of the game in some way that makes it equally likely for Player
A or Player B to win.
One potential change is to change the sums that each players wins with. Here's
one possible solution:
• Player A wins when the sum is 2, 3, 4, 7, 10, 11, or 12.
• Player B wins when the sum is 5, 6, 8, or 9.
It may seem surprising that this is a fair game, but with this change each player
will win one-half (18/36) of the time.
7. Another way to solve this problem is to look at a probability table for the sum of
the two dice. This representation can be quite useful, since it gives us a complete
description of the probabilities for the different values of the sum of two dice,
independent of the rules of the game.
Using the table of the sums, ask participants to complete the probability table:
Sum Frequency Probability
2
1
1/36
3
2
2/36
4
3
3/36
5
4
4/36
6
5
5/36
7
6
6/36
8
5
5/36
9
4
4/36
10
3
3/36
11
2
2/36
12
1
1/36
8. Use the probability table you completed to determine the probability that Player A
will win the game. Recall that Player A wins if the sum is 2, 3, 4, 10, 11, or 12.
This can be found by adding the probabilities of the sums that Player A will win
with:
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Algebra 1 Summer Institute 2014
1/36 + 2/36 + 3/36 + 3/36 + 2/36 + 1/36 = 12/36
Player A wins with probability 12/36, or one-third of the time.
9. If we know the probability that Player A wins, how could you use it to determine
the probability that Player B wins without adding the remaining values in the
table? (Slide 5)
Since one of the two players has to win, the sum of both probabilities -- that of
Player A winning and that of Player B winning -- is 1. So a faster way to find the
probability that Player B wins is to subtract the probability that Player A wins
(12/36) from 1:
1 - (12/36) = (36/36) - (12/36) = 24/36
Player B wins with probability 24/36, or two-thirds of the time.
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